FUNCTIONS CONTINUOUS ON TWICE DIFFERENTIABLE CURVES, DISCONTINUOUS ON LARGE SETS

摘要

We provide a simple construction of a function F: K~2 → R discontinuous on a perfect set P, while having continuous restrictions F|C for all twice differentiable curves C. In particular, F is separately continuous and linearly continuous. While it has been known that the projection n[P] of any such set P onto a straight line must be meager, our construction allows π[P] to have arbitrarily large measure. In particular, P can have arbitrarily large 1-Hausdorff measure, which is the best possible result in this direction, since any such P has Hausdorff dimension at most 1.